It s just the area of a rectangular element, dx by dz, in the xz plane. In polar it would have been rdrdθ, etc.Marcin H said:
So what is the general procedure for finding ds? I feel like I struggle determining ds for problems.haruspex said:
It depends on your coordinate system. With general coordinates, ξ, η say, you consider the product dξdη. In general, the area enclosed by the points (ξ,η), (ξ+dξ,η), (ξ+dξ,η+dη), (ξ,η+dη) might have area equal to dξdη. To make the right area you may need to multiply by a Jacobian.Marcin H said:
A flux integral is a type of line integral that measures the flow of a vector field through a surface. It is used in physics and mathematics to calculate the amount of a quantity passing through a given surface.
To find ds for line integrals, you first need to parameterize the curve or surface you are integrating over. This means expressing the curve or surface as a function of one or more variables. Then, you can use the formula ds = √(dx² + dy² + dz²) for curves in three dimensions or ds = √(dx² + dy²) for curves in two dimensions.
The general formula for a flux integral is ∫∫(F · n)ds, where F is the vector field, n is the unit normal vector to the surface, and ds is the differential of the surface area. This formula is also known as the surface integral of F.
Flux integrals have many applications in physics and engineering. They are used to calculate the electric and magnetic fields in electromagnetic theory, the amount of fluid passing through a surface in fluid dynamics, and the flow of heat through a surface in thermodynamics.
One limitation of flux integrals is that they can only be used to calculate the flux of a vector field through a closed surface. They cannot be used for open surfaces or surfaces with holes. Additionally, the calculation of flux integrals can be complex and time-consuming, requiring knowledge of multivariable calculus and vector analysis.